Schaum's Outline of Discrete Mathematics, Third Edition (Schaum's Outlines)

CHAP. 7] PROBABILITY 127

Theorems on Finite Probability Spaces

The following theorem follows directly from the fact that the probability of an event is the sum of the
probabilities of its points.

Theorem 7.1: The probability functionPdefined on the class of all events in a finite probability space has the
following properties:

[P 1 ] For every eventA,0≤P (A)≤1.

[P 2 ] P(S)=1.

[P 3 ] If eventsAandBare mutually exclusive, thenP(A∪B)=P (A)+P(B).

The next theorem formalizes our intuition that ifpis the probability that an eventEoccurs, then 1−pis
the probability thatEdoes not occur. (That is, if we hit a targetp= 1 /3 of the times, then we miss the target
1 −p= 2 /3 of the times.)

Theorem 7.2: LetAbe any event. ThenP(Ac)= 1 −P (A).

The following theorem (proved in Problem 7.13) follows directly from Theorem 7.1.

Theorem 7.3: Consider the empty set
and any eventsAandB. Then:

(i) P(
)=0.

(ii) P(A\B)=P (A)−P(A∩B).

(iii) IfA⊆B, thenP (A)≤P(B).

Observe that Property[P 3 ]in Theorem 7.1 gives the probability of the union of events in the case that the
events are disjoint. The general formula (proved in Problem 7.14) is called the Addition Principle. Specifically:

Theorem 7.4 (Addition Principle): For any eventsAandB,

P(A∪B)=P (A)+P(B)−P(A∩B)

EXAMPLE 7.6 Suppose a student is selected at random from 100 students where 30 are taking mathematics,
20 are taking chemistry, and 10 are taking mathematics and chemistry. Find the probabilitypthat the student is
taking mathematics or chemistry.
LetM ={students taking mathematics}andC={students taking chemistry}. Since the space is equi-
probable,

P(M)=

30

100

=

3

10

, P (C)=

20

100

=

1

5

,P(MandC)=P(M∩C)=

10

100

=

1

10

Thus, by the Addition Principle (Theorem 7.4),

p=P(MorC)=P(M∪C)=P(M)+P(C)−P(M∩C)=

3

10

+

1

5

−

1

10

=

2

5

7.4Conditional Probability

SupposeEis an event in a sample spaceSwithP(E) >0. The probability that an eventAoccurs onceE
has occurred or, specifically, theconditional probability ofAgivenE. writtenP(A|E), is defined as follows:

P(A|E)=

P(A∩E)

P(E)

As pictured in the Venn diagram in Fig. 7-3,P(A|E)measures, in a certain sense, the relative probability ofA
with respect to the reduced spaceE.